Calculus 8th Edition

Alternating series test: Suppose that we have series $\Sigma a_n$, such that $a_{n}=(-1)^{n}b_n$ or $a_{n}=(-1)^{n+1}b_n$, where $b_n\geq 0$ for all $n$. Then if the following two condition are satisfied the series is convergent. 1. $\lim\limits_{n \to \infty}b_{n}=0$ 2. $b_{n}$ is a decreasing sequence. In the given problem, $b_{n}=\frac{lnn}{\sqrt n}$ 1. $b_{n}=\frac{lnn}{\sqrt n}$, is a decreasing sequence. 2. $\lim\limits_{n \to \infty}b_{n}=\lim\limits_{n \to \infty}\frac{lnn}{\sqrt n}$ $=\lim\limits_{n \to \infty} \frac{2\sqrt n}{n}$ $=0$ Hence, the series is convergent by AST.